A series solution and a fast algorithm for the inversion of the spherical mean Radon transform

An explicit series solution is proposed for the inversion of the spherical mean Radon transform. Such an inversion is required in problems of thermo- and photo- acoustic tomography. Closed-form inversion formulae are currently known only for the case when the centers of the integration spheres lie on a sphere surrounding the support of the unknown function, or on certain unbounded surfaces. Our approach results in an explicit series solution for any closed measuring surface surrounding a region for which the eigenfunctions of the Dirichlet Laplacian are explicitly known - such as, for example, cube, finite cylinder, half-sphere etc. In addition, we present a fast reconstruction algorithm applicable in the case when the detectors (the centers of the integration spheres) lie on a surface of a cube. This algorithm reconsrtucts 3-D images thousands times faster than backprojection-type methods

SAR Image Formation via Subapertures and 2D Backprojection

Radar imaging requires the use of wide bandwidth and a long coherent processing interval, resulting in range and Doppler migration throughout the observation period. This migration must be compensated in order to properly image a scene of interest at full resolution and there are many available algorithms having various strengths and weaknesses. Here, a subaperture-based imaging algorithm is proposed, which first forms range-Doppler (RD) images from slow-time sub-intervals, and then coherently integrates over the resulting coarse-resolution RD maps to produce a full resolution SAR image. A two-dimensional backprojection-style approach is used to perform distortion-free integration of these RD maps. This technique benefits from many of the same benefits as traditional backprojection; however, the architecture of the algorithm is chosen such that several steps are shared with typical target detection algorithms. These steps are chosen such that no compromises need to be made to data quality, allowing for high quality imaging while also preserving data for implementation of detection algorithms. Additionally, the algorithm benefits from computational savings that make it an excellent imaging algorithm for implementation in a simultaneous SAR-GMTI architecture

Fast OPED algorithm for reconstruction of images from Radon data

A fast implementation of the OPED algorithm, a reconstruction algorithm for Radon data introduced recently, is proposed and tested. The new implementation uses FFT for discrete sine transform and an interpolation step. The convergence of the fast implementation is proved under the condition that the function is mildly smooth. The numerical test shows that the accuracy of the OPED algorithm changes little when the fast implementation is used.Comment: 13 page

Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators

The Radon transform and its adjoint, the back-projection operator, can both be expressed as convolutions in log-polar coordinates. Hence, fast algorithms for the application of the operators can be constructed by using FFT, if data is resampled at log-polar coordinates. Radon data is typically measured on an equally spaced grid in polar coordinates, and reconstructions are represented (as images) in Cartesian coordinates. Therefore, in addition to FFT, several steps of interpolation have to be conducted in order to apply the Radon transform and the back-projection operator by means of convolutions. Both the interpolation and the FFT operations can be efficiently implemented on Graphical Processor Units (GPUs). For the interpolation, it is possible to make use of the fact that linear interpolation is hard-wired on GPUs, meaning that it has the same computational cost as direct memory access. Cubic order interpolation schemes can be constructed by combining linear interpolation steps which provides important computation speedup. We provide details about how the Radon transform and the back-projection can be implemented efficiently as convolution operators on GPUs. For large data sizes, speedups of about 10 times are obtained in relation to the computational times of other software packages based on GPU implementations of the Radon transform and the back-projection operator. Moreover, speedups of more than a 1000 times are obtained against the CPU-implementations provided in the MATLAB image processing toolbox